Accelerated Failure Time
Previously, we assumed the hazards were proportional; that is, the covariates multiplied the baseline hazard function:
\[
\begin{aligned}
h(T=t|X=x)
&\stackrel{\text{def}}{=} p(T=t|X=x,T \ge t)\\
&= {\lambda}(t|X=0)\cdot \text{exp}{\left\{\eta(x)\right\}}\\
&= {\lambda}(t|X=0)\cdot \theta(x)\\
&= {\lambda}_0(t)\cdot \theta(x)
\end{aligned}
\]
and correspondingly,
\[
\begin{aligned}
{\Lambda}(t|x)
&= \theta(x){\Lambda}_0(t)\\
\text{S}(t|x)
&= \text{exp}{\left\{-{\Lambda}(t|x)\right\}}\\
&= \text{exp}{\left\{-\theta(x)\cdot {\Lambda}_0(t)\right\}}\\
&= {\left(\text{exp}{\left\{- {\Lambda}_0(t)\right\}}\right)}^{\theta(x)}\\
&= {\left(\text{S}_0(t)\right)}^{\theta(x)}\\
\end{aligned}
\]
An alternative modeling assumption would be \[\text{S}(t|X=x)=\text{S}_0(t\cdot \theta(x))\] where \(\theta(x)=\text{exp}{\left\{\eta(x)\right\}}\), \(\eta(x) =\beta_1x_1+\cdots+\beta_px_p\), and \(\text{S}_0(t)=\text{P}(T\ge t|X=0)\) is the base survival function.
Then
\[
\begin{aligned}
\text{E}[T|X=x]
&= \int_{t=0}^{\infty} \text{S}(t|x)dt\\
&= \int_{t=0}^{\infty} \text{S}_0(t\cdot \theta(x))dt\\
&= \int_{u=0}^{\infty} \text{S}_0(u)du \cdot \theta(x)^{-1}\\
&= \theta(x)^{-1} \cdot \int_{u=0}^{\infty} \text{S}_0(u)du\\
&= \theta(x)^{-1} \cdot \text{E}[T|X=0]\\
\end{aligned}
\] So the mean of \(T\) given \(X=x\) is the baseline mean divided by \(\theta(x) = \text{exp}{\left\{\eta(x)\right\}}\).
This modeling strategy is called an accelerated failure time model, because covariates cause uniform acceleration (or slowing) of failure times.
Additionally:
\[
\begin{aligned}
{\Lambda}(t|x) &= {\Lambda}_0(\theta(x)\cdot t)\\
{\lambda}(t|x) &= \theta(x) \cdot {\lambda}_0(\theta(x)\cdot t)
\end{aligned}
\]
If the base distribution is exponential with parameter \(\lambda\) then
\[
\begin{aligned}
\text{S}(t|x)
&= \text{exp}{\left\{-\lambda \cdot t \theta(x)\right\}}\\
&= [\text{exp}{\left\{-\lambda t\right\}}]^{\theta(x)}\\
\end{aligned}
\]
which is an exponential model with base hazard multiplied by \(\theta(x)\), which is also the proportional hazards model.
In terms of the log survival time \(Y=\log{T}\) the model can be written as
\[
\begin{aligned}
Y&=\alpha-\eta+W\\
\alpha&= -\log{\lambda}
\end{aligned}
\]
where \(W\) has the extreme value distribution. The estimated parameter \(\lambda\) is the intercept and the other coefficients are those of \(\eta\), which will be the opposite sign of those for coxph.
For a Weibull distribution, the hazard function and the survival function are
\[
\begin{aligned}
{\lambda}(t)&=\lambda p (\lambda t)^{p-1}\\
\text{S}(t)&=e^{-(\lambda t)^p}
\end{aligned}
\]
We can construct a proportional hazards model by using a linear predictor \(\eta_i\) without constant term and letting \(\theta_i=e^{\eta_i}\) we have
\[
\begin{aligned}
{\lambda}(t)&=\lambda p (\lambda t)^{p-1}\theta_i
\end{aligned}
\]
A distribution with \({\lambda}(t)=\lambda p (\lambda t)^{p-1}\theta_i\) is a Weibull distribution with parameters \(\lambda^*=\lambda \theta_i^{1/p}\) and \(p\) so the survival function is
\[
\begin{aligned}
S^*(t)&=e^{-(\lambda^* t)^p}\\
&=e^{-(\lambda \theta^{1/p} t)^p}\\
&= \text{S}(t\theta^{1/p})
\end{aligned}
\]
so this is also an accelerated failure time model.
In terms of the log survival time \(Y=\log{T}\) the model can be written as
\[
\begin{aligned}
Y&=\alpha-\sigma\eta+\sigma W\\
\alpha&= -\log{\lambda}\\
\sigma &= 1/p
\end{aligned}
\]
where \(W\) has the extreme value distribution. The estimated parameter \(\lambda\) is the intercept and the other coefficients are those of \(\eta\), which will be the opposite sign of those for coxph.
These AFT models are log-linear, meaning that the linear predictor has a log link. The exponential and the Weibull are the only log-linear models that are simultaneously proportional hazards models. Other parametric distributions can be used for survival regression either as a proportional hazards model or as an accelerated failure time model.
